Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini

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Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini Istituto Nazionale per la Fisica della Materia Research and Development Center on Bose-Einstein Condensation Dipartimento di Fisica – Università di Trento BEC CNR-INFM meeting 2-3 May 2006

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Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini Istituto Nazionale per la Fisica della Materia Research and Development Center on Bose-Einstein Condensation Dipartimento di Fisica – Universit à di Trento. BEC CNR-INFM meeting 2-3 May 2006. - PowerPoint PPT Presentation

Transcript of Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini

Page 1: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Quantum Monte Carlo methodsapplied to ultracold gases

Stefano Giorgini

Istituto Nazionale per la Fisica della Materia Research and Development Center on

Bose-Einstein CondensationDipartimento di Fisica – Università di Trento

BEC CNR-INFM meeting 2-3 May 2006

Page 2: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

QMC simulations have become an important tool in the study of dilute ultracold gases

• Critical phenomena

Shift of Tc in 3D Grüter et al. (´97), Holzmann and Krauth (´99), Kashurnikov et al. (´01)

Kosterlitz-Thouless Tc in 2D Prokof’ev et al. (´01)

• Low dimensions

Large scattering length in 1D and 2D Trento (´04 - ´05)

• Quantum phase transitions in optical latticesBose-Hubbard model in harmonic traps Batrouni et al. (´02)

• Strongly correlated fermions

BCS-BEC crossover Carlson et al. (´03), Trento (´04 - ´05)

Thermodynamics and Tc at unitarity Bulgac et al. (´06), Burovski et al. (´06)

Page 3: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Continuous-space QMC methods

Zero temperature• Solution of the many-body Schrödinger equation

Variational Monte Carlo Based on variational principle

energy upper bound

Diffusion Monte Carlo exact method for the ground state of Bose systems

Fixed-node Diffusion Monte Carlo (fermions and excited states) exact for a given nodal surface energy upper bound

Finite temperature• Partition function of quantum many-body system

Path Integral Monte Carlo exact method for Bose systems

function trial where TTT

TT HE

Page 4: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Low dimensions + large scattering length

Page 5: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

1D Hamiltonian

if g1D large and negative (na1D<<1) metastable gas-like state of hard-rods of size a1D

N

i jiijD

iD zg

zmH

112

22

1 )(2

21

222

)1(1

6 D

HR

namn

NE

at na1D 0.35 the inverse compressibility vanishesgas-like state rapidly disappearsforming clusters

1

323

2

1

2

1 03.1122

aa

maa

mag DD

DD

g1D>0 Lieb-Liniger Hamiltonian (1963)

g1D<0 ground-state is a cluster state

(McGuire 1964)

Olshanii (1998)

Page 6: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Correlations are stronger than in the Tonks-Girardeau gas

(Super-Tonks regime)

Peak in staticstructure factor

Power-law decay in OBDM

Breathing mode inharmonic traps

mean field

TG

Page 7: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Equation of state of a 2D Bose gas

)/1ln(2

22

2

D

MF

nan

mNE

Universality and beyond mean-field effects

• hard disk• soft disk• zero-range

for zero-range potential mc2=0 at na2D

20.04onset of instability for cluster formation

Page 8: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

BCS-BEC crossover in a Fermi gas at T=0

-1/kFa

BCSBEC

Page 9: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

...)(

615

1281

18

52// 2/3

3 mFmFF

b akakNE

BEC regime: gas of molecules [mass 2m - density n/2 – scattering length am]

am=0.6 a (four-body calculation of Petrov et al.)am=0.62(1) a (best fit to FN-DMC)

Equation of state

beyond mean-field effects

confirmed by study of collective modes (Grimm)

Page 10: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Frequency of radial mode (Innsbruck)

Mean-field equation of state

QMCequation of state

Page 11: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Momentum distribution

Condensate fraction

JILA in traps

2/130 )(

38

1 mmann

Page 12: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Static structure factor (Trento + Paris ENS collaboration)

( can be measured in Bragg scattering experiments)

at large momentumtransfer

kF k 1/acrossover from S(k)=2 free moleculestoS(k)=1 free atoms

Page 13: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

New projects:

• Unitary Fermi gas in an optical lattice (G. Astrakharchik + Barcelona)

d=1/q=/2 lattice spacing

Filling 1: one fermion of each spin component per site (Zürich)

Superfluid-insulator transition

single-band Hubbard Hamiltonian is inadequate

)(sin)(sin)(sin2

)( 22222

qzqyqxmq

sVext r

Page 14: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

superfluid fraction condensate fraction

s

Page 15: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

S=1 S=20

Page 16: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

• Bose gas at finite temperature (S. Pilati + Barcelona)

Equation of state and universality

T Tc T Tc

Page 17: Quantum Monte Carlo methods applied to ultracold gases  Stefano Giorgini

Pair-correlation function and bunching effect

Temperature dependence of condensate fraction and superfluid density

(+ N. Prokof’ev’s help on implemention of worm-algorithm)

T = 0.5 Tc